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Theorem toponcom 22300
Description: If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
toponcom ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽)) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐾))

Proof of Theorem toponcom
StepHypRef Expression
1 toponuni 22286 . . . 4 (𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽) β†’ βˆͺ 𝐽 = βˆͺ 𝐾)
21eqcomd 2739 . . 3 (𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽) β†’ βˆͺ 𝐾 = βˆͺ 𝐽)
32anim2i 618 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽)) β†’ (𝐽 ∈ Top ∧ βˆͺ 𝐾 = βˆͺ 𝐽))
4 istopon 22284 . 2 (𝐽 ∈ (TopOnβ€˜βˆͺ 𝐾) ↔ (𝐽 ∈ Top ∧ βˆͺ 𝐾 = βˆͺ 𝐽))
53, 4sylibr 233 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽)) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐾))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆͺ cuni 4869  β€˜cfv 6500  Topctop 22265  TopOnctopon 22282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-topon 22283
This theorem is referenced by:  toponcomb  22301  kgencn3  22932
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